Formal Proof Chain of the σ-Calculus

We provide a full formal derivation of the σ-Calculus: starting from the directed-tension postulate we prove the emergence of Einstein–Hilbert gravity, global unitarity, singularity theorems, a stable Friedmann cosmology and, via a Lyapunov C-function, the universal RG attraction to the Planck fixed point (G,ℏ,c), including mesh-independent convergence and gauge-field extensions. Updates Appendix B. The Block–Laplacian Theorem in the σ–Calculus (new section added to the Appendix VI) Appendix B. Adds the Lyapunov method, which is useful for obtaining stability certificates (aerospace standards and ISO 26262 spectral-proof compliance). Added for completeness: proof of essential self-adjointness of H and K.Notably, the Lyapunov C-function used in the RG analysis is defined independently of any energy or action functional, relying solely on the combinatorial and spectral properties of the discrete curvature field. To fix: Numbering (LaTeX)To fix: Remove 14.1 and keep 14.3 (two versions of the same topic).Note: If any proof component appears overly compact, a detailed or stepwise-expanded version (“exhaustive proof”) can be provided on request. Please indicate the specific lemma, axiom, or derivation line of interest.